Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A solution to the energy minimization problem constrained by a density function


Author: Kanya Ishizaka
Journal: Math. Comp. 86 (2017), 275-314
MSC (2010): Primary 46N10, 52C35; Secondary 49N45, 26A51
DOI: https://doi.org/10.1090/mcom/3136
Published electronically: April 26, 2016
MathSciNet review: 3557800
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a new solution to the problem of determining an energy integral which has a unique minimum at a given Borel probability measure on a compact metric space. For a continuous kernel, we show that there exists a unique weight function such that the given measure is an equilibrium measure with respect to the kernel multiplied by the weight function. The weight function is determined as a unique fixed point of a functional operator. Moreover, if the kernel satisfies the energy principle on the space, then the given measure achieves a unique minimum of the energy integral with respect to the weighted kernel. In order to obtain a kernel satisfying the energy principle on Euclidean subspaces, we improve the condition shown by Gneiting for a defining function of a kernel to belong to the Mittal-Berman-Gneiting class. By using the obtained condition, we show related results for the energy with the kernel. Finally, we present practical examples of distributing a finite number of points that are constrained by a density function.


References [Enhancements On Off] (What's this?)

  • [1] R. Askey, Radial characteristic functions, Tech. Report No. 1262, Math. Research Center, University of Wisconsin-Madison, 1973.
  • [2] Robert B. Ash, Probability and measure theory, 2nd ed., Harcourt/Academic Press, Burlington, MA, 2000. With contributions by Catherine Doléans-Dade. MR 1810041 (2001j:28001)
  • [3] Simeon M. Berman, A class of isotropic distributions in $ {\bf R}^{n}$ and their characteristic functions, Pacific J. Math. 78 (1978), no. 1, 1-9. MR 513278 (80b:60027)
  • [4] S. V. Borodachov, D. P. Hardin, and E. B. Saff, Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1559-1580. MR 2357705 (2009a:49071), https://doi.org/10.1090/S0002-9947-07-04416-9
  • [5] Johann S. Brauchart, Douglas P. Hardin, and Edward B. Saff, Discrete energy asymptotics on a Riemannian circle, Unif. Distrib. Theory 7 (2012), no. 2, 77-108. MR 3052946
  • [6] Steven B. Damelin, On bounds for diffusion, discrepancy and fill distance metrics, Principal manifolds for data visualization and dimension reduction, Lect. Notes Comput. Sci. Eng., vol. 58, Springer, Berlin, 2008, pp. 261-270. MR 2447230 (2009m:65047), https://doi.org/10.1007/978-3-540-73750-6_11
  • [7] S. B. Damelin, A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of Euclidean space, Numer. Algorithms 48 (2008), no. 1-3, 213-235. MR 2413284 (2009d:41002), https://doi.org/10.1007/s11075-008-9187-6
  • [8] Bent Fuglede, On the theory of potentials in locally compact spaces, Acta Math. 103 (1960), 139-215. MR 0117453 (22 #8232)
  • [9] Tilmann Gneiting, Radial positive definite functions generated by Euclid's hat, J. Multivariate Anal. 69 (1999), no. 1, 88-119. MR 1701408 (2000g:60022), https://doi.org/10.1006/jmva.1998.1800
  • [10] Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer Monographs in Mathematics, Springer, Heidelberg, 2014. MR 3244285
  • [11] N. Goloshchapova, M. Malamud, and V. Zastavnyi, Radial positive definite functions and spectral theory of the Schrödinger operators with point interactions, Math. Nachr. 285 (2012), no. 14-15, 1839-1859. MR 2988008, https://doi.org/10.1002/mana.201100132
  • [12] B. I. Golubov, On Abel-Poisson type and Riesz means, Anal. Math. 7 (1981), no. 3, 161-184 (English, with Russian summary). MR 635483 (83b:42015), https://doi.org/10.1007/BF01908520
  • [13] M. Götz, On the Riesz energy of measures, J. Approx. Theory 122 (2003), no. 1, 62-78. MR 1976125 (2004b:42068), https://doi.org/10.1016/S0021-9045(03)00031-5
  • [14] Manuel Gräf, Daniel Potts, and Gabriele Steidl, Quadrature errors, discrepancies, and their relations to halftoning on the torus and the sphere, SIAM J. Sci. Comput. 34 (2012), no. 5, A2760-A2791. MR 3023725, https://doi.org/10.1137/100814731
  • [15] Douglas P. Hardin, Amos P. Kendall, and Edward B. Saff, Polarization optimality of equally spaced points on the circle for discrete potentials, Discrete Comput. Geom. 50 (2013), no. 1, 236-243. MR 3070548, https://doi.org/10.1007/s00454-013-9502-4
  • [16] Lester L. Helms, Potential theory, Universitext, Springer-Verlag London, Ltd., London, 2009. MR 2526019 (2011a:31001)
  • [17] Kanya Ishizaka, A minimum energy condition of 1-dimensional periodic sphere packing, JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 3, Article 80, 8 pp. (electronic). MR 2164321 (2006d:05045)
  • [18] K. Ishizaka, Apparatus and image processing apparatus for estimating distribution of objects and defining distribution of objects, Japanese Patent Laid-open Publication (in Japanese), No. 2007-189427, 2007.
  • [19] Kanya Ishizaka, A local minimum energy condition of hexagonal circle packing, JIPAM. J. Inequal. Pure Appl. Math. 9 (2008), no. 3, Article 66, 26. MR 2443739 (2009m:26011)
  • [20] Kanya Ishizaka, New spatial measure for dispersed-dot halftoning assuring good point distribution in any density, IEEE Trans. Image Process. 18 (2009), no. 9, 2030-2047. MR 2750932, https://doi.org/10.1109/TIP.2009.2022443
  • [21] Kanya Ishizaka, On power sums of convex functions in local minimum energy problem, J. Math. Inequal. 6 (2012), no. 2, 253-271. MR 2977785, https://doi.org/10.7153/jmi-06-26
  • [22] Kanya Ishizaka, Weak$ ^*$-convergence to minimum energy measure and dispersed-dot halftoning, SIAM J. Imaging Sci. 7 (2014), no. 2, 1035-1079. MR 3209722, https://doi.org/10.1137/130941894
  • [23] W. A. Kirk and B. Sims (ed.), Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271 (2003b:47002)
  • [24] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073 (2007a:34002)
  • [25] N. S. Landkof, Foundations of modern potential theory, translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. MR 0350027 (50 #2520)
  • [26] Vladimir Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, second, revised and augmented edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530 (2012a:46056)
  • [27] Yashaswini Mittal, A class of isotropic covariance functions, Pacific J. Math. 64 (1976), no. 2, 517-538. MR 0426122 (54 #14068)
  • [28] N. Ninomiya, Potential Theory (in Japanese) [Revised publication of the original, 1969], Kyoritsu Shuppan, Tokyo, 2009.
  • [29] Erich Novak and Henryk Woźniakowski, Tractability of multivariate problems. Volume II: Standard information for functionals, EMS Tracts in Mathematics, vol. 12, European Mathematical Society (EMS), Zürich, 2010. MR 2676032 (2011h:46093)
  • [30] Makoto Ohtsuka, On potentials in locally compact spaces, J. Sci. Hiroshima Univ. Ser. A-I Math. 25 (1961), 135-352. MR 0180695 (31 #4926)
  • [31] Keith B. Oldham and Jerome Spanier, The fractional calculus: Theory and applications of differentiation and integration to arbitrary order, Mathematics in Science and Engineering, vol. 111, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. With an annotated chronological bibliography by Bertram Ross. MR 0361633 (50 #14078)
  • [32] Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778 (99h:31001)
  • [33] T. Teuber, G. Steidl, P. Gwosdek, C. Schmaltz, and J. Weickert, Dithering by differences of convex functions, SIAM J. Imaging Sci. 4 (2011), no. 1, 79-108. MR 2765671 (2012f:65234), https://doi.org/10.1137/100790197
  • [34] R. E. Williamson, Multiply monotone functions and their Laplace transforms, Duke Math. J. 23 (1956), 189-207. MR 0077581 (17,1061d)
  • [35] J. Yeh, Real analysis: Theory of measure and integration, 2nd ed., World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. MR 2250344 (2007i:28001)
  • [36] Victor P. Zastavnyi, On positive definiteness of some functions, J. Multivariate Anal. 73 (2000), no. 1, 55-81. MR 1766121 (2002b:42017), https://doi.org/10.1006/jmva.1999.1864

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 46N10, 52C35, 49N45, 26A51

Retrieve articles in all journals with MSC (2010): 46N10, 52C35, 49N45, 26A51


Additional Information

Kanya Ishizaka
Affiliation: Key Technology Laboratory, Research & Technology Group, Fuji Xerox Co., Ltd., 430 Sakai, Nakai-machi, Ashigarakami-gun, Kanagawa, 259-0157, Japan
Email: Kanya.Ishizaka@fujixerox.co.jp

DOI: https://doi.org/10.1090/mcom/3136
Keywords: Minimum energy state, optimal distribution, density constraint, positive definite kernel, energy principle, halftoning
Received by editor(s): November 22, 2012
Received by editor(s) in revised form: July 10, 2015
Published electronically: April 26, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society